Integrand size = 46, antiderivative size = 27 \[ \int \frac {-2 x+e^x \left (-x+x^2\right )+e^x \left (-x-x^2\right ) \log \left (e^{-x} x\right )-4 \log \left (x^2\right )}{x} \, dx=3-2 x-e^x x \log \left (e^{-x} x\right )-\log ^2\left (x^2\right ) \]
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Time = 0.03 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.96, number of steps used = 6, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {14, 2326, 2338} \[ \int \frac {-2 x+e^x \left (-x+x^2\right )+e^x \left (-x-x^2\right ) \log \left (e^{-x} x\right )-4 \log \left (x^2\right )}{x} \, dx=-\log ^2\left (x^2\right )-2 x-e^x x \log \left (e^{-x} x\right ) \]
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Rule 14
Rule 2326
Rule 2338
Rubi steps \begin{align*} \text {integral}& = \int \left (-e^x \left (1-x+\log \left (e^{-x} x\right )+x \log \left (e^{-x} x\right )\right )-\frac {2 \left (x+2 \log \left (x^2\right )\right )}{x}\right ) \, dx \\ & = -\left (2 \int \frac {x+2 \log \left (x^2\right )}{x} \, dx\right )-\int e^x \left (1-x+\log \left (e^{-x} x\right )+x \log \left (e^{-x} x\right )\right ) \, dx \\ & = -e^x x \log \left (e^{-x} x\right )-2 \int \left (1+\frac {2 \log \left (x^2\right )}{x}\right ) \, dx \\ & = -2 x-e^x x \log \left (e^{-x} x\right )-4 \int \frac {\log \left (x^2\right )}{x} \, dx \\ & = -2 x-e^x x \log \left (e^{-x} x\right )-\log ^2\left (x^2\right ) \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.96 \[ \int \frac {-2 x+e^x \left (-x+x^2\right )+e^x \left (-x-x^2\right ) \log \left (e^{-x} x\right )-4 \log \left (x^2\right )}{x} \, dx=-2 x-e^x x \log \left (e^{-x} x\right )-\log ^2\left (x^2\right ) \]
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Time = 0.24 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.93
method | result | size |
default | \(-2 x -{\mathrm e}^{x} \ln \left (x \,{\mathrm e}^{-x}\right ) x -\ln \left (x^{2}\right )^{2}\) | \(25\) |
parallelrisch | \(-2 x -{\mathrm e}^{x} \ln \left (x \,{\mathrm e}^{-x}\right ) x -\ln \left (x^{2}\right )^{2}\) | \(25\) |
parts | \(-2 x -{\mathrm e}^{x} \ln \left (x \,{\mathrm e}^{-x}\right ) x -\ln \left (x^{2}\right )^{2}\) | \(25\) |
risch | \(\ln \left ({\mathrm e}^{x}\right ) x \,{\mathrm e}^{x}-4 \ln \left (x \right )^{2}-x \,{\mathrm e}^{x} \ln \left (x \right )-2 x +2 i \ln \left (x \right ) \pi \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )-4 i \ln \left (x \right ) \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{2}+2 i \ln \left (x \right ) \pi \operatorname {csgn}\left (i x^{2}\right )^{3}+\frac {i {\mathrm e}^{x} \pi x \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i {\mathrm e}^{-x}\right ) \operatorname {csgn}\left (i x \,{\mathrm e}^{-x}\right )}{2}-\frac {i {\mathrm e}^{x} \pi x \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x \,{\mathrm e}^{-x}\right )^{2}}{2}-\frac {i {\mathrm e}^{x} \pi x \,\operatorname {csgn}\left (i {\mathrm e}^{-x}\right ) \operatorname {csgn}\left (i x \,{\mathrm e}^{-x}\right )^{2}}{2}+\frac {i {\mathrm e}^{x} \pi x \operatorname {csgn}\left (i x \,{\mathrm e}^{-x}\right )^{3}}{2}\) | \(176\) |
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Time = 0.25 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int \frac {-2 x+e^x \left (-x+x^2\right )+e^x \left (-x-x^2\right ) \log \left (e^{-x} x\right )-4 \log \left (x^2\right )}{x} \, dx=x^{2} e^{x} - \frac {1}{2} \, x e^{x} \log \left (x^{2}\right ) - \log \left (x^{2}\right )^{2} - 2 \, x \]
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Time = 0.14 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.96 \[ \int \frac {-2 x+e^x \left (-x+x^2\right )+e^x \left (-x-x^2\right ) \log \left (e^{-x} x\right )-4 \log \left (x^2\right )}{x} \, dx=- 2 x + \frac {\left (2 x^{2} - x \log {\left (x^{2} \right )}\right ) e^{x}}{2} - \log {\left (x^{2} \right )}^{2} \]
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Time = 0.22 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.41 \[ \int \frac {-2 x+e^x \left (-x+x^2\right )+e^x \left (-x-x^2\right ) \log \left (e^{-x} x\right )-4 \log \left (x^2\right )}{x} \, dx={\left (x^{2} - x \log \left (x\right ) - x + 2\right )} e^{x} + {\left (x - 1\right )} e^{x} - \log \left (x^{2}\right )^{2} - 2 \, x - e^{x} \]
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Time = 0.28 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.93 \[ \int \frac {-2 x+e^x \left (-x+x^2\right )+e^x \left (-x-x^2\right ) \log \left (e^{-x} x\right )-4 \log \left (x^2\right )}{x} \, dx=x^{2} e^{x} - x e^{x} \log \left (x\right ) - \log \left (x^{2}\right )^{2} - 2 \, x \]
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Time = 12.72 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.26 \[ \int \frac {-2 x+e^x \left (-x+x^2\right )+e^x \left (-x-x^2\right ) \log \left (e^{-x} x\right )-4 \log \left (x^2\right )}{x} \, dx=-{\ln \left (x^2\right )}^2-2\,x-{\mathrm {e}}^x\,\left (x+x\,\ln \left (x\right )-x^2-2\right )+{\mathrm {e}}^x\,\left (x-2\right ) \]
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